A notion of embedding appropriate to higher-order syntax is described. This provides a representation of annotated formulae in terms of the difference between pairs of formulae. We define substitution and unification for such annotated terms. Using this representation of annotated terms, the proof search guidance technique of rippling can be extended to higher-order theorems. We illustrate this by several examples based on an implementation of these ideas in Prolog.

We present a new approach to introducing an extensional propositional equality in Intensional Type Theory. Our construction is based on the observation that there is a sound, intensional setoid model in Intensional Type theory with a proof-irrelevant universe of propositions and -rules for - and -types. The Type Theory corresponding to this model is decidable, has no irreducible constants and permits large eliminations, which are essential for universes. Keywords. Type Theory, categorical models. 1. Introduction and Summary In Intensional Type Theory (see e.g. [11]) we differentiate between a decidable definitional equality (which we denote by =) and a propositional equality type (Id ( ; ) for any given type ) which requires proof. Typing only depends on definitional equality and hence is decidable. In Intensional Type Theory the type corresponding to the principle of extensionality Ext x2:(x) f;g2(x2:(x)) ( x2 Id (x) (f(x); g(x))) ! Id x2:(x) (f; g) is not...

One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand. Acknowledgement. The following persons provided help in various ways. Erik Barendsen, Jon Barwise, Johan van Benthem, Andreas Blass, Olivier Danvy, Wil Dekkers, Marko van Eekelen, Sol Feferman, Andrzej Filinski, Twan Laan, Jan Kuper, Pierre Lescanne, Hans Mooij, Robert Maron, Rinus Plasmeijer, Randy Pollack, Kristoffer Rose, Richard Shore, Rick Statman and Simon Thompson. Partial support came from the European HCM project Typed lambda calculus (CHRXCT-92-0046), the Esprit Working Group Types (21900) and the Dutch NWO project WINST (612-316-607). 1. Introduction This paper is written to honor Church's gr...

this paper we investigate the interaction of notational definitions with algorithms for testing equality and unification. We propose a syntactic criterion on definitions which avoids their expansion in many cases without losing soundness or completeness with respect to fi ffi-conversion. Our setting is the dependently typed -calculus [HHP93], but, with minor modifications, our results should apply to richer type theories and logics. The question when definitions need to be expanded is surprisingly subtle and of great practical importance. Most algorithms for equality and unification rely on decomposing a problem

Formal proofs in mathematics and computer science are being studied because these objects can be verified by a very simple computer program. An important open problem is whether these formal proofs can be generated with an effort not much greater than writing a mathematical paper in, say, L A T E X. Modern systems for proof-development make the formalization of reasoning relatively easy. Formalizing computations such that the results can be used in formal proofs is not immediate. In this paper it is shown how to obtain formal proofs of statements like Prime(61) in the context of Peano arithmetic or (x + 1)(x + 1) = x 2 + 2x + 1 in the context of rings. It is hoped that the method will help bridge the gap between the efficient systems of computer algebra and the reliable systems of proof-development. 1. The problem Usual mathematics is informal but precise. One speaks about informal rigor. Formal mathematics on the other hand consists of definitions, statements and proo...

We present a personal view and strategy for algorithm-supported mathematical theory exploration and draw some conclusions for the desirable functionality of future mathematical software systems. The main points of emphasis are: The use of schemes for bottom-up mathematical invention, the algorithmic generation of conjectures from failing proofs for top-down mathematical invention, and the possibility to program new reasoners within the logic on which the reasoners work ("meta-programming").

. This paper investigates the semantics of mathematical concepts in a type theoretic framework with coercive subtyping. The typetheoretic analysis provides a formal semantic basis in the design and implementation of Mathematical Vernacular (MV), a natural language suitable for interactive development of mathematics with the support of the current theorem proving technology. The idea of semantic well-formedness in mathematical language is motivated with examples. A formal system based on a notion of conceptual category is then presented, showing how type checking supports our notion of well-formedness. The power of this system is then extended by incorporating a notion of subcategory, using ideas from a more general theory of coercive subtyping, which provides the mechanisms for modelling conventional abbreviations in mathematics. Finally, we outline how this formal work can be used in an implementation of MV. 1 Introduction By mathematical vernacular (MV), we mean a mathematical and n...

We present a simple and effective methodology for equational reasoning in proof checkers. The method is based on a two-level approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with inductive and oracle types. The potential of our two-level approach is illustrated by some examples developed in Lego.

The Tecton Proof System is an experimental tool for constructing proofs of first order logic formulas and of program specifications expressed using formulas in Hoare's axiomatic proof formalism. It is designed to make interactive proof construction easier than with previous proof tools, by maintaining multiple proof attempts internally in a structured form called a proof forest; displaying them in an easy to comprehend form, using a combination of tabular formats, graphical representations, and hypertext links; and automating substantial parts of proofs through rewriting, induction, case analysis, and generalization inference mechanisms, along with a linear arithmetic decision procedure. Further development of the system is planned as part of an overall framework aimed at supporting the kind of abstractions and specializations necessary for building libraries of generic software and hardware components. Partially supported by National Science Foundation Grants CCR--8906678...

Abstract. This paper considers the use of dependent types to capture information about dynamic resource usage in a static type system. Dependent types allow us to give (explicit) proofs of properties with a program; we present a dependently typed core language ��, and define a framework within this language for representing size metrics and their properties. We give several examples of size bounded programs within this framework and show that we can construct proofs of their size bounds within ��. We further show how the framework handles recursive higher order functions and sum types, and contrast our system with previous work based on sized types. 1